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Bimetric gravity or bigravity refers to two different class of theories. The first class of theories rely on modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one.[1][2] The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent, enabling models with a variable speed of light.

If the two metrics are dynamical and interact, a first possibility implies two graviton modes, one massive and one massless; such bimetric theories are then closely related to massive gravity.[3] Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995)[4][5][6] or Mordehai Milgrom with relativistic extensions of Modified Newtonian Dynamics (MOND).[7] More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity.[8] Though none has been shown to account for physical observations more accurately or more consistently than the theory of general relativity, Rosen's theory has been shown to be inconsistent with observations of the Hulse–Taylor binary pulsar.[5] Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.[9][10]

On the contrary, the second class of bimetric gravity theories does not rely on massive gravitons and does not modify Newton's law, but instead describes the universe as a manifold having two coupled Riemannian metrics, where matter populating the two sectors interact through gravitation (and antigravitation if the topology and the Newtonian approximation considered introduce negative mass and negative energy states in cosmology as an alternative to dark matter and dark energy). Some of these cosmological models also use a variable speed of light in the high energy density state of the radiation-dominated era of the universe, challenging the inflation hypothesis.[11][12][13][14][15]

Rosen's bigravity (1940 to 1989)

In general relativity (GR), it is assumed that the distance between two points in spacetime is given by the metric tensor. Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum.

In 1940, Rosen[1][2] proposed that at each point of space-time, there is a Euclidean metric tensor \( \gamma _{ij} \) in addition to the Riemannian metric tensor \( g_{ij} \). Thus at each point of space-time there are two metrics:

\( {\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}} \)
\( {\displaystyle d\sigma ^{2}=\gamma _{ij}dx^{i}dx^{j}} \)

The first metric tensor, \( g_{ij} \), describes the geometry of space-time and thus the gravitational field. The second metric tensor, \( \gamma _{ij} \), refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from\( g_{ij} \) and \( \gamma _{ij} \) are denoted by \( \{_{{jk}}^{{i}}\} \) and \( \Gamma _{{jk}}^{{i}} \) respectively.

Since the difference of two connections is a tensor, one can define the tensor field \( {\displaystyle \Delta _{jk}^{i}} \) given by:

\( {\displaystyle \Delta _{jk}^{i}=\{_{jk}^{i}\}-\Gamma _{jk}^{i}} \) (1)

Two kinds of covariant differentiation then arise: g-differentiation based on \( g_{ij} \) (denoted by a semicolon, e.g. \( {\displaystyle X_{;a}}) \), and covariant differentiation based on \( \gamma _{ij} \) (denoted by a slash, e.g. \( {\displaystyle X_{/a}}) \). Ordinary partial derivatives are represented by a comma (e.g. \( {\displaystyle X_{,a}}) \) . Let \( {\displaystyle R_{ijk}^{h}} \) and \( {\displaystyle P_{ijk}^{h}} \) be the Riemann curvature tensors calculated from \( g_{ij} \) and \( \gamma _{ij} \), respectively. In the above approach the curvature tensor \( {\displaystyle P_{ijk}^{h}} \) is zero, since \(\gamma _{ij} \) is the flat space-time metric.

A straightforward calculation yields the Riemann curvature tensor

\( {\displaystyle R_{ijk}^{h}=P_{ijk}^{h}-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}=-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}} \)

Each term on the right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing {:} by Δ {\displaystyle \Delta } \Delta and ordinary differentiation by covariant \( \gamma \) -differentiation, \( {\sqrt {-g}} \) by \( {\displaystyle {\sqrt {\tfrac {g}{\gamma }}}} \), integration measure \( d^{{4}}x \) by \( {\displaystyle {\sqrt {-\gamma }}\,d^{4}x} \), where \( {\displaystyle g=\det(g_{ij})} \), \( {\displaystyle \gamma =\det(\gamma _{ij})} \) and \( d^{{4}}x=dx^{{1}}dx^{{2}}dx^{{3}}dx^{{4}} \). Having once introduced \( \gamma _{ij} \) into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up other field equations other than Einstein's. It is possible that some of these will be more satisfactory for the description of nature.

The geodesic equation in bimetric relativity (BR) takes the form

\( {\displaystyle {\frac {d^{2}x^{i}}{ds^{2}}}+\Gamma _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}+\Delta _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}=0} \) (2)

It is seen from equations (1) and (2) that Γ {\displaystyle \Gamma } \Gamma can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.

Being the quantity \( \Delta \) a tensor, it is independent of any coordinate system and hence may be regarded as describing the permanent gravitational field.

Rosen (1973) has found BR satisfying the covariance and equivalence principle. In 1966, Rosen showed that the introduction of the space metric into the framework of general relativity not only enables one to get the energy momentum density tensor of the gravitational field, but also enables one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are

\( {\displaystyle K_{j}^{i}=N_{j}^{i}-{\frac {1}{2}}\delta _{j}^{i}N=-8\pi \kappa T_{j}^{i}} \) (3)

where

\( {\displaystyle N_{j}^{i}={\frac {1}{2}}\gamma ^{\alpha \beta }(g^{hi}g_{hj/\alpha })_{/\beta }} \)

or

\( {\displaystyle {\begin{aligned}N_{j}^{i}&={\frac {1}{2}}\gamma ^{\alpha \beta }\left\{\left(g^{hi}g_{hj,\alpha }\right)_{,\beta }-\left(g^{hi}g_{mj}\Gamma _{h\alpha }^{m}\right)_{,\beta }-\gamma ^{\alpha \beta }\left(\Gamma _{j\alpha }^{i}\right)_{,\beta }+\Gamma _{\lambda \beta }^{i}\left[g^{h\lambda }g_{hj,\alpha }-g^{h\lambda }g_{mj}\Gamma _{h\alpha }^{m}-\Gamma _{j\alpha }^{\lambda }\right]-\right.\\&\qquad \Gamma _{j\beta }^{\lambda }\left[g^{hi}g_{h\lambda ,\alpha }-g^{hi}g_{m\lambda }\Gamma _{h\alpha }^{m}-\Gamma _{\lambda \alpha }^{i}\right]+\Gamma _{\alpha \beta }^{\lambda }\left.\left[g^{hi}g_{hj,\lambda }-g^{hi}g_{mj}\Gamma _{h\lambda }^{m}-\Gamma _{j\lambda }^{i}\right]\right\}\end{aligned}}} \)

with

\( {\displaystyle N=g^{ij}N_{ij}} \), \( {\displaystyle \kappa ={\sqrt {\frac {g}{\gamma }}}} \)

and \( T_{{j}}^{{i}} \) is the energy-momentum tensor.

The variational principle also leads to the relation

\( {\displaystyle T_{j;i}^{i}=0}. \)

Hence from (3)

\( {\displaystyle K_{j;i}^{i}=0}, \)

which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to \( g_{{ij}} \) .

Rosen continued improving his bimetric gravity theory with additional publications in 1978[16] and 1980,[17] in which he made an attempt "to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe." In 1985[18] Rosen tried again to remove singularities and pseudo-tensors from General Relativity. Twice in 1989 with publications in March[19] and November[20] Rosen further developed his concept of elementary particles in a bimetric field of General Relativity.

It is found that the BR and GR theories differ in the following cases:

propagation of electromagnetic waves
the external field of a high density star
the behaviour of intense gravitational waves propagating through a strong static gravitational field.

The predictions of gravitational radiation in Rosen's theory have been shown since 1992 to be in conflict with observations of the Hulse–Taylor binary pulsar.[5]
Massive bigravity
Main article: Massive gravity

Since 2010 there has been renewed interest in bigravity after the development by Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT) of a healthy theory of massive gravity.[21] Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric \( g_{\mu \nu } \) can only be written down with the help of a second metric, as the only nonderivative term that can be written using one metric is a cosmological constant. In the dRGT theory, a nondynamical "reference metric" \( f_{{\mu \nu }} \) is introduced, and the interaction terms are built out of the matrix square root of \( g^{{-1}}f. \)

In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric an Einstein–Hilbert term, in which case \( f_{{\mu \nu }} \) is not chosen but instead evolves dynamically in response to \( g_{\mu \nu } \) and possibly matter. This massive bigravity was introduced by Fawad Hassan and Rachel Rosen as an extension of dRGT massive gravity.[3][22]

The dRGT theory is crucial to developing a theory with two dynamical metrics because general bimetric theories are plagued by the Boulware–Deser ghost, a possible sixth polarization for a massive graviton.[23] The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the kinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free.[3]

The action for the ghost-free massive bigravity is given by[24]

\( S = -\frac{M_g^2}{2}\int d^4x \sqrt{-g}R(g )-\frac{M_f^2}{2}\int d^4x \sqrt{-f}R(f) + m^2M_g^2\int d^4x\sqrt{-g}\displaystyle\sum_{n=0}^4\beta_ne_n(\mathbb{X}) + \int d^4x\sqrt{-g}\mathcal{L}_\mathrm{m}(g,\Phi_i). \)

As in standard general relativity, the metric \( g_{\mu \nu } \) has an Einstein–Hilbert kinetic term proportional to the Ricci scalar R(g) and a minimal coupling to the matter Lagrangian \( {\mathcal {L}}_{{\mathrm {m}}} \), with \( \Phi _{i} \) representing all of the matter fields, such as those of the Standard Model. An Einstein–Hilbert term is also given for \( f_{{\mu \nu }} \). Each metric has its own Planck mass, denoted \( M_g \) and \( M_{f} \) respectively. The interaction potential is the same as in dRGT massive gravity. The \( \beta _{i} \) are dimensionless coupling constants and m {\displaystyle m} m (or specifically \( \beta_i^{1/2}m) \)is related to the mass of the massive graviton. This theory propagates seven degrees of freedom, corresponding to a massless graviton and a massive graviton (although the massive and massless states do not align with either of the metrics).

The interaction potential is built out of the elementary symmetric polynomials \( e_{n} \) of the eigenvalues of the matrices \( {\mathbb K}={\mathbb I}-{\sqrt {g^{{-1}}f}} \) or \( {\mathbb X}={\sqrt {g^{{-1}}f}} \), parametrized by dimensionless coupling constants \( \alpha _{i} \) or \( \beta _{i} \), respectively. Here \( {\sqrt {g^{{-1}}f}} \) is the matrix square root of the matrix \( g^{{-1}}f \). Written in index notation, \( \mathbb {X} \) is defined by the relation

\( X^{\mu }{}_{\alpha }X^{\alpha }{}_{\nu }=g^{{\mu \alpha }}f_{{\nu \alpha }}. \)

The \( e_{n} \) can be written directly in terms of \( \mathbb {X} \) as

\( {\begin{aligned}e_{0}({\mathbb X})&=1,\\e_{1}({\mathbb X})&=[{\mathbb X}],\\e_{2}({\mathbb X})&={\frac 12}\left([{\mathbb X}]^{2}-[{\mathbb X}^{2}]\right),\\e_{3}({\mathbb X})&={\frac 16}\left([{\mathbb X}]^{3}-3[{\mathbb X}][{\mathbb X}^{2}]+2[{\mathbb X}^{3}]\right),\\e_{4}({\mathbb X})&=\operatorname {det}{\mathbb X},\end{aligned}} \)

where brackets indicate a trace, \( [{\mathbb X}]\equiv X^{\mu }{}_{\mu } \) . It is the particular antisymmetric combination of terms in each of the \( e_{n} \)which is responsible for rendering the Boulware–Deser ghost nondynamical.
See also

Alternatives to general relativity
DGP model
Scalar–tensor theory

References

Rosen, Nathan (1940), "General Relativity and Flat Space. I", Phys. Rev., 57 (2): 147–150, Bibcode:1940PhRv...57..147R, doi:10.1103/PhysRev.57.147
Rosen, Nathan (1940), "General Relativity and Flat Space. II", Phys. Rev., 57 (2): 150, Bibcode:1940PhRv...57..150R, doi:10.1103/PhysRev.57.150
Hassan, S.F.; Rosen, Rachel A. (2012). "Bimetric Gravity from Ghost-free Massive Gravity". JHEP. 1202 (2): 126. arXiv:1109.3515. Bibcode:2012JHEP...02..126H. doi:10.1007/JHEP02(2012)126.
Rosen, Nathan (1973), "A bi-metric Theory of Gravitation", Gen. Rel. Grav., 4 (6): 435–447, Bibcode:1973GReGr...4..435R, doi:10.1007/BF01215403
Will, Clifford (1992). "The renaissance of general relativity". In Davies, Paul (ed.). The New Physics. Cambridge University Press. p. 18. ISBN 9780521438315. OCLC 824636830. "One interesting by-product of this was the knocking down of the Rosen bimetric theory of gravity, which hitherto was in agreement with solar system experiments. The theory turned out to make radically different predictions for gravitational wave energy loss than general relativity, and was in severe disagreement with the observations."
"Nathan Rosen — The Man and His Life-Work", Technion.ac.il, 2011, web: Technion-rosen.
Milgrom, M. (2009). "Bimetric MOND gravity". Physical Review D. 80 (12). arXiv:0912.0790. doi:10.1103/PhysRevD.80.123536.
Zyga, Lisa (21 September 2017). "Gravitational waves may oscillate, just like neutrinos". Phys.org. Omicron Technology Limited.
Akrami, Yashar; Koivisto, Tomi S.; Sandstad, Marit (2013). "Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality". JHEP. 1303 (3): 099. arXiv:1209.0457. Bibcode:2013JHEP...03..099A. doi:10.1007/JHEP03(2013)099.
Akrami, Yashar; Hassan, S.F.; Könnig, Frank; Schmidt-May, Angnis; Solomon, Adam R. (2015). "Bimetric gravity is cosmologically viable". Physics Letters B. 748: 37–44. arXiv:1503.07521. Bibcode:2015PhLB..748...37A. doi:10.1016/j.physletb.2015.06.062.
Henry-Couannier, F. (30 April 2005). "Discrete symmetries and general relativity, the dark side of gravity". International Journal of Modern Physics A. 20 (11): 2341–2345. arXiv:gr-qc/0410055. Bibcode:2005IJMPA..20.2341H. doi:10.1142/S0217751X05024602.
Hossenfelder, S. (15 August 2008). "A Bi-Metric Theory with Exchange Symmetry". Physical Review D. 78 (4): 044015. arXiv:0807.2838. Bibcode:2008PhRvD..78d4015H. doi:10.1103/PhysRevD.78.044015.
Hossenfelder, Sabine (June 2009). Antigravitation. 17th International Conference on Supersymmetry and the Unification of Fundamental Interactions. Boston: American Institute of Physics. arXiv:0909.3456. doi:10.1063/1.3327545.
Petit, J.-P.; d'Agostini, G. (10 November 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X.
O'Dowd, Matt (7 February 2019). "Sound Waves from the Beginning of Time". PBS Space Time. PBS. 16 minutes in. Retrieved 8 February 2019. "An alternate model that how negative mass might behave: in so-called 'bimetric gravity' you can have positive and negative masses, but each is described by its own set of Einstein field equations. That's kinda like having 'parallel spacetimes', one with positive and one with negative masses, which can still interact gravitationally. In these models, like masses attract and opposite masses repel… and you don't get the crazy 'runaway motion' that occurs if you put both positive and negative masses in the same spacetime. So no perpetual motion machines… It can also be used to explain dark energy and dark matter. An example is the Janus model of Jean-Pierre Petit. This is a much more sophisticated model than the one by Jamie Farnes. It is however just as speculative."
Rosen, Nathan (April 1978). "Bimetric gravitation theory on a cosmological basis". General Relativity and Gravitation. 9 (4): 339–351. Bibcode:1978GReGr...9..339R. doi:10.1007/BF00760426.
Rosen, Nathan (October 1980). "General relativity with a background metric". Foundations of Physics. 10 (9–10): 673–704. Bibcode:1980FoPh...10..673R. doi:10.1007/BF00708416.
Rosen, Nathan (October 1985). "Localization of gravitational energy". Foundations of Physics. 15 (10): 997–1008. Bibcode:1985FoPh...15..997R. doi:10.1007/BF00732842.
Rosen, Nathen (March 1989). "Elementary particles in bimetric general relativity". Foundations of Physics. 19 (3): 339–348. Bibcode:1989FoPh...19..339R. doi:10.1007/BF00734563.
Rosen, Nathan (November 1989). "Elementary particles in bimetric general relativity. II". Foundations of Physics. 19 (11): 1337–1344. Bibcode:1989FoPh...19.1337R. doi:10.1007/BF00732755.
de Rham, Claudia; Gabadadze, Gregory; Tolley, Andrew J. (2011). "Resummation of Massive Gravity". Physical Review Letters 106 (23): 231101. arXiv:1011.1232. Bibcode:2011PhRvL.106w1101D. doi:10.1103/PhysRevLett.106.231101. PMID 21770493.
Merali, Zeeya (2013-09-10). "Fat gravity particle gives clues to dark energy". Nature News. Retrieved 2019-01-23.
Boulware, David G.; Deser, Stanley (1972). "Can gravitation have a finite range?" (PDF). Physical Review D6 (12): 3368–3382. Bibcode:1972PhRvD...6.3368B. doi:10.1103/PhysRevD.6.3368.

Hassan, S.F.; Rosen, Rachel A. (2011). "On Non-Linear Actions for Massive Gravity". JHEP. 1107 (7): 009. arXiv:1103.6055. Bibcode:2011JHEP...07..009H. doi:10.1007/JHEP07(2011)009.

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